2 * Copyright (c) 2016 Thomas Pornin <pornin@bolet.org>
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29 br_i31_muladd_small(uint32_t *x, uint32_t z, const uint32_t *m)
34 uint32_t a0, a1, b0, hi, g, q, tb;
39 * We can test on the modulus bit length since we accept to
50 lo = (x[1] << 31) | z;
51 x[1] = br_rem(hi, lo, m[1]);
54 mlen = (m_bitlen + 31) >> 5;
55 mblr = (unsigned)m_bitlen & 31;
58 * Principle: we estimate the quotient (x*2^31+z)/m by
59 * doing a 64/32 division with the high words.
63 * a = (w*a0 + a1) * w^N + a2
72 * I.e. the two top words of a are a0:a1, the top word of b is
73 * b0, we ensured that b0 is "full" (high bit set), and a is
74 * such that the quotient q = a/b fits on one word (0 <= q < w).
76 * If a = b*q + r (with 0 <= r < q), we can estimate q by
77 * doing an Euclidean division on the top words:
78 * a0*w+a1 = b0*u + v (with 0 <= v < b0)
79 * Then the following holds:
86 memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
91 a0 = ((x[mlen] << (31 - mblr)) | (x[mlen - 1] >> mblr))
93 memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
95 a1 = ((x[mlen] << (31 - mblr)) | (x[mlen - 1] >> mblr))
97 b0 = ((m[mlen] << (31 - mblr)) | (m[mlen - 1] >> mblr))
102 * We estimate a divisor q. If the quotient returned by br_div()
104 * -- If a0 == b0 then g == 0; we want q = 0x7FFFFFFF.
106 * -- if g == 0 then we set q = 0;
107 * -- otherwise, we set q = g - 1.
108 * The properties described above then ensure that the true
109 * quotient is q-1, q or q+1.
111 * Take care that a0, a1 and b0 are 31-bit words, not 32-bit. We
112 * must adjust the parameters to br_div() accordingly.
114 g = br_div(a0 >> 1, a1 | (a0 << 31), b0);
115 q = MUX(EQ(a0, b0), 0x7FFFFFFF, MUX(EQ(g, 0), 0, g - 1));
118 * We subtract q*m from x (with the extra high word of value 'hi').
119 * Since q may be off by 1 (in either direction), we may have to
120 * add or subtract m afterwards.
122 * The 'tb' flag will be true (1) at the end of the loop if the
123 * result is greater than or equal to the modulus (not counting
124 * 'hi' or the carry).
128 for (u = 1; u <= mlen; u ++) {
129 uint32_t mw, zw, xw, nxw;
133 zl = MUL31(mw, q) + cc;
134 cc = (uint32_t)(zl >> 31);
135 zw = (uint32_t)zl & (uint32_t)0x7FFFFFFF;
141 tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
145 * If we underestimated q, then either cc < hi (one extra bit
146 * beyond the top array word), or cc == hi and tb is true (no
147 * extra bit, but the result is not lower than the modulus). In
148 * these cases we must subtract m once.
150 * Otherwise, we may have overestimated, which will show as
151 * cc > hi (thus a negative result). Correction is adding m once.
154 under = ~over & (tb | LT(cc, hi));
155 br_i31_add(x, m, over);
156 br_i31_sub(x, m, under);